73.26.4 problem 36.2 (d)

Internal problem ID [15665]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (d)
Date solved : Thursday, March 13, 2025 at 06:14:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (-x^{2}+2\right ) y^{\prime \prime }+\left (4 x^{2}+5 x \right ) y^{\prime }+\left (x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 47

Order:=6; 
ode:=x^2*(-x^2+2)*diff(diff(y(x),x),x)+(4*x^2+5*x)*diff(y(x),x)+(x^2+1)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} \left (1+4 x +\frac {1}{6} x^{2}-\frac {14}{45} x^{3}+\frac {209}{2520} x^{4}-\frac {823}{28350} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x}+\frac {c_{2} \left (1+\frac {2}{3} x -\frac {19}{120} x^{2}+\frac {1}{180} x^{3}-\frac {23}{51840} x^{4}+\frac {557}{1425600} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 86

ode=x^2*(2-x^2)*D[y[x],{x,2}]+(5*x+4*x^2)*D[y[x],x]+(1+x^2)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to \frac {c_1 \left (\frac {557 x^5}{1425600}-\frac {23 x^4}{51840}+\frac {x^3}{180}-\frac {19 x^2}{120}+\frac {2 x}{3}+1\right )}{\sqrt {x}}+\frac {c_2 \left (-\frac {823 x^5}{28350}+\frac {209 x^4}{2520}-\frac {14 x^3}{45}+\frac {x^2}{6}+4 x+1\right )}{x} \]

✓ Sympy. Time used: 1.236 (sec). Leaf size: 15

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*(2 - x**2)*Derivative(y(x), (x, 2)) + (x**2 + 1)*y(x) + (4*x**2 + 5*x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = \frac {C_{2}}{\sqrt {x}} + \frac {C_{1}}{x} + O\left (x^{6}\right ) \]